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Theorem f1orn 6840
Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
f1orn (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◑𝐹))

Proof of Theorem f1orn
StepHypRef Expression
1 dff1o2 6835 . 2 (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◑𝐹 ∧ ran 𝐹 = ran 𝐹))
2 eqid 2732 . . 3 ran 𝐹 = ran 𝐹
3 df-3an 1089 . . 3 ((𝐹 Fn 𝐴 ∧ Fun ◑𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ Fun ◑𝐹) ∧ ran 𝐹 = ran 𝐹))
42, 3mpbiran2 708 . 2 ((𝐹 Fn 𝐴 ∧ Fun ◑𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ (𝐹 Fn 𝐴 ∧ Fun ◑𝐹))
51, 4bitri 274 1 (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◑𝐹))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  β—‘ccnv 5674  ran crn 5676  Fun wfun 6534   Fn wfn 6535  β€“1-1-ontoβ†’wf1o 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3954  df-ss 3964  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547
This theorem is referenced by:  f1f1orn  6841  infdifsn  9648  efopnlem2  26156  cycpmcl  32262
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